Try it online! Takes input as an 8x8 board of :s with the knights marked with Ns, with a digit for the number of turns on the next line (it makes no sense to have more than 9 turns, but if you insist I can support it for an extra byte). Note that the output contains additional white space. Edit: Saved 45 bytes thanks to @MartinEnder. Explanation: The first stage converts the number of turns to unary, but using tab characters, so that they don't get matched later by accident, while the second stage adds some spaces to the right of the board to stop the regexes from wrapping across the edge. The third stage replaces all Ns and :s that are a knight's move away from an N with an n while the fourth stage deletes any remaining Ns, changes the ns to Ns, and subtracts 1 from the move count. This repeats until the move count is zero.

Input and output are 1...64 based.
Takes a vector of position using the 1...64 notation. Maps it to a 1:576 notation, that is a super-board made of nine boards.
In this notation, at each iteration, each knight should be able to move by +/- 22,26,47,49
Return future positions back in the 1...64 notation, excluding those that fall off the central board. The TIO example displays the result using an 8x8 matrix.

Link 1 (The chessboard)

Link 2 (Move generation)

“¦Ʋƈ2’D_2ṡ2+€µẎ
“¦Ʋƈ2’ = the number 103414301
D = converted into a list of digits
_2 = subtract two from each element
ṡ2 = overlapping pairs
+€ = add to the list of squares
µ = Make sure the next part isn't treated as a right argument
Ẏ = Tighten the list (Reducing the depth by one)

Link 3 (square checking)

ÇƓ¡f1£Q
ÇƓ¡ = Repeat link #2 the requested amount of times
f1£ = Remove everything not a member of link #1 (not on the chess board)
Q = Make sure squares don't appear more than once

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